# Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime

July 06, 2009

The discovery of the radiation properties of black holes prompted the search
for a natural candidate quantum ground state for a massless scalar field theory
on Schwarzschild spacetime, here considered in the Eddington-Finkelstein
representation. Among the several available proposals in the literature, an
important physical role is played by the so-called Unruh state which is
supposed to be appropriate to capture the physics of a black hole formed by
spherically symmetric collapsing matter. Within this respect, we shall consider
a massless Klein-Gordon field and we shall rigorously and globally construct
such state, that is on the algebra of Weyl observables localised in the union
of the static external region, the future event horizon and the non-static
black hole region. Eventually, out of a careful use of microlocal techniques,
we prove that the built state fulfils, where defined, the so-called Hadamard
condition; hence, it is perturbatively stable, in other words realizing the
natural candidate with which one could study purely quantum phenomena such as
the role of the back reaction of Hawking's radiation. From a geometrical point
of view, we shall make a profitable use of a bulk-to-boundary reconstruction
technique which carefully exploits the Killing horizon structure as well as the
conformal asymptotic behaviour of the underlying background. From an analytical
point of view, our tools will range from Hormander's theorem on propagation of
singularities, results on the role of passive states, and a detailed use of the
recently discovered peeling behaviour of the solutions of the wave equation in
Schwarzschild spacetime.

Keywords:

Unruh state, QFT on curved spacetimes